Finite Group Databasebeta

← #18006173

Subgroups — #18006173

Order 4736.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924569C211yesmaximal, characteristic, complemented
2C2×Q16×D37C211yesmaximal, characteristic, complemented
3#14558196C211yesmaximal, characteristic, complemented
4#14558193C211yesmaximal, characteristic
5#14557926C211yesmaximal, characteristic, complemented
6#14557763C211yesmaximal, characteristic, complemented
7#14557758C211yesmaximal, characteristic, complemented
8#14557742C211yesmaximal, characteristic, complemented
9#14557693C211yesmaximal, characteristic, complemented
10#14557690C211yesmaximal, characteristic, complemented
11#14557688C211yesmaximal, characteristic, complemented
12#14557673C211yesmaximal, characteristic, complemented
13#14557639C211yesmaximal, characteristic, complemented
14#14557602C211yesmaximal, characteristic, complemented
15C37×#470C211yesmaximal, characteristic, complemented
16C2×#10596962C2211yescharacteristic
17C22×Dic74C2211yescharacteristic
18C2×Q8×D37C2211yescharacteristic
19C2×Q8×D37C2211yescharacteristic
20#11924569—24no—
21C2×Dic148C2211yescharacteristic
22C2×Dic148C2211yescharacteristic
23C2×#10596942C2211yescharacteristic
24C2×#10596942C2211yescharacteristic
25C2×#10596942C2211yescharacteristic
26Q16×D37—24no—
27C2×#10596930C2211yescharacteristic
28C2×#10596930C2211yescharacteristic
29C2×#10596941C2211yescharacteristic
30C2×C8×D37C2211yescharacteristic
31#11924535—21no—
32#11924534C2211yescharacteristic
33#11924525C2211yescharacteristic
34#11924522—21no—
35#11924516C2211yescharacteristic
36#11924514—21no—
37#11924514—21no—
38#11924513—21no—
39#11924511C2211yescharacteristic
40#11924509—21no—
41#11924501C2211yescharacteristic
42#11924501C2211yescharacteristic
43#11924498—21no—
44C4×C22⋊D37C2211yescharacteristic
45#11924496—21no—
46#11924465C2211yescharacteristic
47#11924459C2211yescharacteristic
48#11924459C2211yescharacteristic
49#11924459C2211yescharacteristic
50#11924456C2211yescharacteristic
51C22⋊#64436C2211yescharacteristic
52#11924448C2211yescharacteristic
53#11924446C2211yescharacteristic
54C2×Q16×C37C2211yescharacteristic
55C2×M16×C37C2211yescharacteristic
56C22⋊Q8×C37C2211yescharacteristic
57C22⋊Q8×C37C2211yescharacteristic
58#106×C37C2211yescharacteristic
59#103×C37C2211yescharacteristic
60#103×C37C2211yescharacteristic
61C2×Q8×C37C2311yescharacteristic
62C2×Q8×C37C2311yescharacteristic
63C22×C4×C37C2311yescharacteristic
64C2×C22⋊D37—21no—
65C2×C22⋊D37D411yescharacteristic
66C22×Dic37—21no—
67Q8×D37—22no—
68Q8×D37—22no—
69Q8×D37—41no—
70Q8×D37—41no—
71#10596962—22no—
72#10596962—41no—
73C2×D148—21no—
74C2×D148C2311yescharacteristic
75C2×C4×D37—21no—
76C2×C4×D37—21no—
77C2×C4×D37—21no—
78C2×C4×D37C2311yescharacteristic
79C2×Dic74—22no—
80C2×Dic74—21no—
81C2×Dic74—21no—
82C2×Dic74—21no—
83C2×Dic74—41no—
84C2×Dic74C2311yescharacteristic
85C2×Dic74C2311yescharacteristic
86C2×Dic74C2311yescharacteristic
87Q16×C37—22no—
88M16×C37—22no—
89C2×C8×C37C2311yescharacteristic
90C2×C8×C37C2311yescharacteristic
91C4⋊C4×C37—21no—
92C4⋊C4×C37—21no—
93C4⋊C4×C37C2311yescharacteristic
94C4⋊C4×C37C2311yescharacteristic
95C22⋊C4×C37—21no—
96C22⋊C4×C37—21no—
97C22⋊Dic37D411yescharacteristic
98#10596942—22no—
99#10596942—22no—
100#10596942—41no—
101#10596941—41no—
102#10596938—21no—
103#10596938—21no—
104#10596938—21no—
105#10596938—21no—
106#10596938D411yescharacteristic
107C4⋊Dic37—21no—
108#10596936—21no—
109#10596936—21no—
110#10596936—21no—
111#10596936—21no—
112#10596936D411yescharacteristic
113C4×Dic37—21no—
114C4×Dic37—21no—
115C4×Dic37C2311yescharacteristic
116C2×#64436C2311yescharacteristic
117C2×#64436C2311yescharacteristic
118Dic148—22no—
119Dic148D414yescomplemented
120#10596930—22no—
121#10596930—22no—
122C8×D37—22no—
123C23×C37C2×D411yescharacteristic
124C22×D37—21no—
125C22×D37C2×D411yescharacteristic
126Q8×C37—21no—
127Q8×C37—21no—
128Q8×C37—21no—
129Q8×C37—21no—
130C2×C4×C37—21no—
131C2×C4×C37—21no—
132C2×C4×C37—21no—
133C2×C4×C37—21no—
134C2×C4×C37—21no—
135C2×C4×C37—21no—
136C2×C4×C37C2×D411yescharacteristic
137C2×C4×C37C2411yescharacteristic
138C22⋊D37—22no—
139C22⋊D37—42no—
140C2×Dic37—21no—
141C2×Dic37—21no—
142C2×Dic37—21no—
143C2×Dic37—42no—
144C2×Dic37C2×D411yescharacteristic
145C2×Dic37C2×D411yescharacteristic
146C2×Dic37C2×D411yescharacteristic
147D148—21no—
148D148—21no—
149D148—42no—
150C4×D37—21no—
151C4×D37—21no—
152C4×D37—42no—
153C4×D37—42no—
154C4×D37—42no—
155Dic74—21no—
156Dic74—21no—
157Dic74—21no—
158Dic74—21no—
159Dic74—42no—
160Dic74—42no—
161Dic74—42no—
162Dic74C2×D412yes—
163Dic74C2×D412yes—
164C8×C37—21no—
165C8×C37C2×D412yes—
166#64436—21no—
167#64436—21no—
168C22×C37—21no—
169C22×C37—21no—
170C22×C37—21no—
171C22×C37C22×D411yescharacteristic
172C2×D37—21no—
173C2×D37—21no—
174C2×D37—41no—
175C2×D37—41no—
176C2×D37—41no—
177C2×D37#13712yescomplemented
178C4×C37—21no—
179C4×C37—41no—
180C4×C37—41no—
181C4×C37—41no—
182C4×C37—41no—
183C4×C37C22×D411yescharacteristic
184C4×C37H(5,𝔽2)11yescharacteristic
185Dic37—22no—
186Dic37—22no—
187Dic37—21no—
188Dic37—21no—
189Dic37—21no—
190Dic37—41no—
191#3270—371nomaximal, Sylow
192C2×C37—41no—
193C2×C37D4211yescharacteristic
194C2×C37C2×#13711yescharacteristic
195C2×C37#57811yescharacteristic
196D37—22no—
197D37—81no—
198C2×#137—371no—
199C22×Q16—371no—
200D4×Q8—371no—
201#548—371no—
202#497—371no—
203#470D3711yescharacteristic, complemented
204#465—371no—
205#464—371no—
206#462—371no—
207#452—371no—
208#451—371no—
209#451—371no—
210#448—371no—
211#441—371no—
212#435—371no—
213C37#327011yescharacteristic, complemented, Sylow
214C2×#41—371no—
215C22×Q8—371no—
216C22×Q8—371no—
217C22×Q8—371no—
218#137—744no—
219C2×Q16—371no—
220C2×Q16—371no—
221C2×Q16—371no—
222C2×Q16—371no—
223C2×Q16—371no—
224C2×Q16—744no—
225C2×Q16C2×D3711yescharacteristic
226C2×SD16—371no—
227C2×SD16—371no—
228C2×SD16—371no—
229C2×M16C2×D3711yescharacteristic
230C22×C8—371no—
231C4⋊Q8—371no—
232C4⋊Q8—741no—
233#124—371no—
234#124—741no—
235C22⋊Q8—371no—
236C22⋊Q8—741no—
237C22⋊Q8—741no—
238C22⋊Q8—741no—
239C22⋊Q8C2×D3711yescharacteristic
240C22⋊Q8C2×D3711yescharacteristic
241#121—371no—
242#121—741no—
243C4×Q8—371no—
244C4×Q8—371no—
245C4×D4—371no—
246C4×D4—741no—
247C4×D4—741no—
248#106C2×D3711yescharacteristic
249C4⋊C8—371no—
250#103—371no—
251#103—371no—
252#103—371no—
253#103C2×D3711yescharacteristic
254#103C2×D3711yescharacteristic
255#102—371no—
256C22⋊C8—371no—
257C22⋊C8—371no—
258#97—371no—
259#41—742no—
260#41—1481no—
261C2×Q8—371no—
262C2×Q8—371no—
263C2×Q8—371no—
264C2×Q8—741no—
265C2×Q8—741no—
266C2×Q8—742no—
267C2×Q8—741no—
268C2×Q8—742no—
269C2×Q8—742no—
270C2×Q8—1481no—
271C2×Q8—1481no—
272C2×Q8—1481no—
273C2×Q8C22×D3711yescharacteristic
274C2×Q8C22×D3711yescharacteristic
275C2×D4—371no—
276C2×D4—371no—
277C2×D4—741no—
278C2×D4—741no—
279C22×C4—371no—
280C22×C4—741no—
281C22×C4—741no—
282C22×C4—741no—
283C22×C4—741no—
284C22×C4C22×D3711yescharacteristic
285Q16—22no—
286Q16—374no—
287Q16—742no—
288Q16—742no—
289Q16—742no—
290Q16—1481no—
291SD16—742no—
292SD16—742no—
293SD16—1481no—
294M16—22no—
295C2×C8—371no—
296C2×C8—371no—
297C2×C8—742no—
298C2×C8C22×D3711yescharacteristic
299C2×C8C22×D3711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—371no—
303C4⋊C4—741no—
304C4⋊C4—741no—
305C4⋊C4—741no—
306C4⋊C4—741no—
307C4⋊C4—741no—
308C4⋊C4C22×D3711yescharacteristic
309C4⋊C4C22×D3711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—371no—
313C22⋊C4—371no—
314C22⋊C4—741no—
315C22⋊C4—741no—
316C22⋊C4—741no—
317C22⋊C4—741no—
318C42—371no—
319C42—741no—
320C42—741no—
321C23—371no—
322C23—741no—
323C23D4×D3711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—372no—
329Q8—372no—
330Q8—741no—
331Q8—741no—
332Q8—741no—
333Q8—741no—
334Q8—1482no—
335Q8—1482no—
336Q8—1482no—
337D4—742no—
338D4—741no—
339D4—741no—
340D4—1482no—
341D4—1482no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—371no—
349C2×C4—371no—
350C2×C4—371no—
351C2×C4—741no—
352C2×C4—741no—
353C2×C4—741no—
354C2×C4—741no—
355C2×C4—741no—
356C2×C4—1482no—
357C2×C4—1482no—
358C2×C4—1482no—
359C2×C4—1482no—
360C2×C4D4×D3711yescharacteristic
361C2×C4C23×D3711yescharacteristic
362C8—21no—
363C8—741no—
364C8—741no—
365C8D4×D3712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—372no—
370C22—741no—
371C22—741no—
372C22—1481no—
373C22—1481no—
374C22—1481no—
375C22C2×D4×D3711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—741no—
382C4—741no—
383C4—741no—
384C4—742no—
385C4—742no—
386C4—1481no—
387C4C2×D4×D3711yescharacteristic
388C4#1192458211yescharacteristic
389C2—41no—
390C2—2961no—
391C2—742no—
392C2#1455816711yescharacteristic
393C2#137×D3711yescharacteristic
394C2#1455834911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#470C2
socleC22×C37C22×D4
prime radical#14558193C2
hypercenter#470D37
hypocenterC37#3270
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#11924569C2semidirect
2. C2×Q16×D37C2semidirect
3. #14558196C2semidirect
5. #14557926C2semidirect
6. #14557763C2semidirect
7. #14557758C2semidirect
8. #14557742C2semidirect
9. #14557693C2semidirect
10. #14557690C2semidirect
11. #14557688C2semidirect
12. #14557673C2semidirect
13. #14557639C2semidirect
14. #14557602C2semidirect
15. C37×#470C2semidirect
119. Dic148D4semidirect
177. C2×D37#137semidirect
203. #470D37semidirect
213. C37#3270semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#119245692
C2×Q16×D372
#145581962
#145581932
#145579262
#145577632
#145577582
#145577422
#145576932
#145576902
#145576882
#145576732
#145576392
#145576022
C37×#4702
#327037